Support vector machines : optimization based theory, algorithms, and extensions


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Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Yarlagadda, Merla- Introduction The support vector machine is a powerful state of algorithm that has both the theoretical foundations and strong regularization properties. The strong generalization properties make Support vector machines able to generalize the model easily to the new data. Support vector machine is a kernel based algorithm which uses quadratic programming that transforms the input data into high dimensional space for finding the global optimum. The banking industry is a key element in any industrial economy.

For the banks to be healthy, they must use marketing campaigns to attract and retain customers. The prediction of the success rate of marketing campaigns can be handled via different models.

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The goal of this paper is to show that effective use of SVM is the best for handling the data with large number of variables where the outcome of the campaign occupies the space in four different quadrants of the dimensional space which cannot be fitted via the linear fashion. The data set consists of number of observations and 25 variables such as like job, education, marital status, contact type, day, month, duration, last contact date, type of campaign, response etc.

For categorical variables, a dummy variable is created with the case values like 0 and 1 to indicate the type of the category. The validation average squared error, misclassification rate, ROC curve and cumulative lift statistics are used to evaluate the performance of the models.

SAGE Reference - Support Vector Machines

Radial Basis and Sigmoid SVM models turned out to be the best models with a validation average squared error of 0. The outcome of success of the campaign mainly depends on the last contact and the status of the previous marketing campaign.


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March and November seem the best months for the marketing of bank campaigns. Figure 1.

Model building Discussion: 1 We might end up with a binary target occupying the non-linear shape in the dimensional space with the dataset having a large number of variables. SVM are the best type of models which maps the data into four dimensional spaces and gives us the unbiased results with more accuracy. Goutam Chakraborty Figure 2. Prediction Accuracy Results Figure 3.

ROC curve of the different models Results References.


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Goutam Chakraborty for his guidance and advice on this project. Best Models Figure 5. Regularization parameter used : 0. Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training data points of any class so-called functional margin , since in general the larger the margin the lower the generalization error of the classifier. Whereas the original problem may be stated in a finite dimensional space, it often happens that in that space the sets to be discriminated are not linearly separable.

For this reason it was proposed that the original finite dimensional space be mapped into a much higher dimensional space presumably making the separation easier in that space. SVM schemes use a mapping into a larger space so that cross products may be computed easily in terms of the variables in the original space making the computational load reasonable. The hyperplanes in the large space are defined as the set of points whose cross product with a vector in that space is constant.

Motivation for Support Vector Machines

In this way the sum of kernels above can be used to measure the relative nearness of each test point to the data points originating in one or the other of the sets to be discriminated. Statistical classification Classifying data is a common task in machine learning. Suppose some given data points each belong to one of two classes, and the goal is to decide which class a new data point will be in. This is called a linear classifier.

There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized.

If such a hyperplane exists, it is known as the maximum-margin hyperplane and the linear classifier it defines is known as a maximum margin classifier.

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These hyperplanes can be described by the equations. Note that if the training data are linearly separable, we can select the two hyperplanes of the margin in a way that there are no points between them and then try to maximize their distance. For simplicity reasons, sometimes it is required that the hyperplane passes through the origin of the coordinate system.

Such hyperplanes are called unbiased , whereas general hyperplanes not necessarily passing through the origin are called biased. The corresponding dual is identical to the dual given above without the equality constraint. Transductive support vector machines extend SVMs in that they could also treat partially labeled data in semi-supervised learning. Formally, a transductive support vector machine is defined by the following primal optimization problem:.

SVMs belong to a family of generalized linear classifiers. They can also be considered a special case of Tikhonov regularization.

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A special property is that they simultaneously minimize the empirical classification error and maximize the geometric margin ; hence they are also known as maximum margin classifiers. In , Corinna Cortes and Vladimir Vapnik suggested a modified maximum margin idea that allows for mislabeled examples. If the penalty function is linear, the optimization problem becomes:.


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One has then to solve the following problem. The key advantage of a linear penalty function is that the slack variables vanish from the dual problem, with the constant C appearing only as an additional constraint on the Lagrange multipliers. Non-linear penalty functions have been used, particularly to reduce the effect of outliers on the classifier, but unless care is taken, the problem becomes non-convex, and thus it is considerably more difficult to find a global solution.

The original optimal hyperplane algorithm proposed by Vladimir Vapnik in was a linear classifier. However, in , Bernhard Boser, Isabelle Guyon and Vapnik suggested a way to create non-linear classifiers by applying the kernel trick originally proposed by Aizerman et al. This allows the algorithm to fit the maximum-margin hyperplane in a transformed feature space. The transformation may be non-linear and the transformed space high dimensional; thus though the classifier is a hyperplane in the high-dimensional feature space, it may be non-linear in the original input space.

If the kernel used is a Gaussian radial basis function also known as RBF, the corresponding feature space is a Hilbert space of infinite dimension. Maximum margin classifiers are well regularization mathematics regularized, so the infinite dimension does not spoil the results.

Support vector machines : optimization based theory, algorithms, and extensions Support vector machines : optimization based theory, algorithms, and extensions
Support vector machines : optimization based theory, algorithms, and extensions Support vector machines : optimization based theory, algorithms, and extensions
Support vector machines : optimization based theory, algorithms, and extensions Support vector machines : optimization based theory, algorithms, and extensions
Support vector machines : optimization based theory, algorithms, and extensions Support vector machines : optimization based theory, algorithms, and extensions
Support vector machines : optimization based theory, algorithms, and extensions Support vector machines : optimization based theory, algorithms, and extensions

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